Diffusive limit of the Boltzmann equation around Rayleigh profile in the half space

Abstract

This paper concerns the diffusive limit of the time evolutionary Boltzmann equation in the half space $\mathbb{T}^2\times\mathbb{R}^+$ for a small Knudsen number $\varepsilon>0$. For boundary conditions in the normal direction, it involves diffuse reflection moving with a tangent velocity proportional to $\varepsilon$ on the wall, whereas the far field is described by a global Maxwellian with zero bulk velocity. The incompressible Navier-Stokes equations, as the corresponding formal fluid dynamic limit, admit a specific time-dependent shearing solution known as the Rayleigh profile, which accounts for the effect of the tangentially moving boundary on the flow at rest in the far field. Using the Hilbert expansion method, for well-prepared initial data we construct the Boltzmann solution around the Rayleigh profile without initial singularity over any finite time interval.